Find the values of the following: cos⁻¹(1/2) + 2sin⁻¹(1/2)
2π/3
Explanation
The question requires evaluating the sum of inverse trigonometric functions using their principal values. The principal branch of cos⁻¹x is [0, π] and that of sin⁻¹x is [-π/2, π/2], as referenced in the context (e.g., question 13). The standard angles where cos θ = 1/2 and sin θ = 1/2 are θ = π/3 and θ = π/6 respectively, both lying within the respective principal ranges. Hence, cos⁻¹(1/2) = π/3 and sin⁻¹(1/2) = π/6. Substituting these gives the final result 2π/3.
Solution Steps
Step 1: Find cos⁻¹(1/2). Since cos(π/3) = 1/2 and π/3 ∈ [0, π] (principal range of cos⁻¹), we have cos⁻¹(1/2) = π/3.
Step 2: Find sin⁻¹(1/2). Since sin(π/6) = 1/2 and π/6 ∈ [-π/2, π/2] (principal range of sin⁻¹), we have sin⁻¹(1/2) = π/6.
Step 3: Compute the expression: cos⁻¹(1/2) + 2 sin⁻¹(1/2) = π/3 + 2 × (π/6) = π/3 + π/3 = 2π/3.